Fujita phenomena in nonlinear pseudo-parabolic system  

作　　者：YANG JinGe CAO Yang ZHENG SiNing 

机构地区：[1]School of Mathematical Sciences, Dalian University of Technology

出　　处：《Science China Mathematics》2014年第3期555-568,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11171048 and 11201047);the Doctor Startup Foundation of Liaoning Province(Grant No.20121025);the Fundamental Research Funds for the Central Universities

摘　　要：This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut-△u-αut=vp,vt-△v-αvt=uqwith p,q 1 and pq>1,where the viscous terms of third order are included.We first find the critical Fujita exponent,and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions.Moreover,time-decay profiles are obtained for the global solutions.It can be found that,diferent from those for the situations of general semilinear heat systems,we have to use distinctive techniques to treat the influence from the viscous terms of the highest order.To fix the non-global solutions,we exploit the test function method,instead of the general Kaplan method for heat systems.To obtain the global solutions,we apply the Lp-Lq technique to establish some uniform Lmtime-decay estimates.In particular,under a suitable classification for the nonlinear parameters and the initial data,various Lmtime-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system.It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing efect to establish the compactness of approximating solutions,which cannot be directly realized here due to absence of the smooth efect in the pseudo-parabolic system.

关 键 词：semilinear pseudo-parabolic system critical Fujita exponent second critical exponent global profile 

分 类 号：O175.26[理学—数学]